Solving Right Triangles: Finding Missing Angles
Use inverse trig functions — arcsin, arccos, arctan — to find a missing angle when two sides of a right triangle are known.
Sometimes you don't know the angle — you know the sides. A ramp has a rise of 3 feet and a run of 20 feet: what is the angle? A cable is attached 50 feet up a wall and anchored 30 feet from the base: at what angle does it meet the ground? Inverse trig functions (arcsin, arccos, arctan) reverse the process, turning a known side ratio back into an angle. This is essential in engineering, robotics, game physics, and any field where geometry is measured.
- Understand what "inverse trig" means and when to use it
- Apply \(\arcsin\), \(\arccos\), and \(\arctan\) to find a missing angle
- Select the correct inverse function based on which two sides are known
- Solve a right triangle completely given two sides
If \(\sin\theta = 0.5\), then \(\theta = \arcsin(0.5) = 30°\). Use inverse trig when two sides are known and you need the angle.
\[ \theta = \arcsin\!\left(\frac{\text{opp}}{\text{hyp}}\right) \qquad \theta = \arccos\!\left(\frac{\text{adj}}{\text{hyp}}\right) \qquad \theta = \arctan\!\left(\frac{\text{opp}}{\text{adj}}\right) \]On a calculator: use the sin⁻¹, cos⁻¹, or tan⁻¹ button (often accessed via 2nd + sin/cos/tan).
Which Inverse Function to Use?
| Sides Known | Inverse Function | Formula |
|---|---|---|
| Opposite + Hypotenuse | arcsin | \(\theta = \sin^{-1}(\text{opp}/\text{hyp})\) |
| Adjacent + Hypotenuse | arccos | \(\theta = \cos^{-1}(\text{adj}/\text{hyp})\) |
| Opposite + Adjacent | arctan | \(\theta = \tan^{-1}(\text{opp}/\text{adj})\) |
A right triangle has opposite = 5 and adjacent = 12. Find angle θ.
\[ \theta = \arctan\!\left(\frac{5}{12}\right) = \arctan(0.4167) \approx \mathbf{22.6°} \]A right triangle has legs a = 9 and b = 12. Find both acute angles and the hypotenuse.
- Hypotenuse: \(c = \sqrt{81+144} = \sqrt{225} = 15\)
- Angle A (opposite a, adjacent b): \(\theta_A = \arctan(9/12) = \arctan(0.75) \approx \mathbf{36.87°}\)
- Angle B: \(90° - 36.87° = \mathbf{53.13°}\) (angles of a triangle sum to 180°)
A loading ramp rises 2.5 feet over a horizontal run of 18 feet. What angle does the ramp make with the ground?
\[ \theta = \arctan\!\left(\frac{2.5}{18}\right) = \arctan(0.1389) \approx \mathbf{7.9°} \]The ramp angle is about 7.9°. ADA regulations require ramps to be ≤ 4.76° for accessible slopes — this ramp is steeper than recommended.
- Opposite = 8, hypotenuse = 17. Find θ using arcsin.
- Adjacent = 6, hypotenuse = 10. Find θ using arccos.
- A 15-foot ladder leans against a wall with its base 6 feet from the wall. What angle does the ladder make with the ground?
▶ Show Answers
- \(\theta = \sin^{-1}(8/17) = \sin^{-1}(0.471) \approx \mathbf{28.1°}\).
- \(\theta = \cos^{-1}(6/10) = \cos^{-1}(0.6) \approx \mathbf{53.1°}\).
- \(\theta = \cos^{-1}(6/15) = \cos^{-1}(0.4) \approx \mathbf{66.4°}\).
- Using sin instead of sin⁻¹: \(\sin(0.5) \approx 0.479\) (not 30°). You need \(\sin^{-1}(0.5) = 30°\). Make sure you're pressing the inverse button.
- Calculator in wrong mode: Inverse trig on a calculator in radian mode gives radians (e.g., 0.524 rad ≈ 30°). If you want degrees, set your calculator to degree mode first.
- Forgetting the second angle: In a right triangle, once you find one acute angle, the other is \(90° - \theta\). Always complete the picture.
- Two sides known, angle unknown → inverse trig function.
- Use \(\arcsin\) (opp/hyp), \(\arccos\) (adj/hyp), or \(\arctan\) (opp/adj).
- The other acute angle is always \(90° - \theta\).
- Always check: all three angles must sum to 180°.
Robot arms need to know the angle of every joint to position a tool precisely. Given the
desired x-y coordinates of the tool tip and the arm segment lengths, inverse kinematics uses
arctan (often the two-argument version, atan2) to compute joint angles. This
is the same math — sides-to-angles — applied to servo motor positioning, drone flight
control, and CNC machine movement.
Calculator Connection
The Right Triangle Solver uses inverse trig internally to find missing angles from any two known sides. The Inverse Trig Functions Calculator gives arcsin, arccos, and arctan for any ratio.
Try it with the Calculator
Apply what you've learned with these tools.
Solving Right Triangles: Finding Missing Angles — Quiz
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